0-7803-7952-7/03/$17.00 ©2003 IEEE DEPSO: Hybrid Particle Swarm with Differential Evolution Operator Wen-Jun Zhang, Xiao-Feng Xie* Institute of Microelectronics, Tsinghua University, Beijing 100084, P. R. China *Email: xiexiaofeng@tsinghua.org.cn Abstract - A hybrid particle swarm with differential evolution operator, termed DEPSO, which provide the bell-shaped mutations with consensus on the population diversity along with the evolution, while keeps the self- organized particle swarm dynamics, is proposed. Then it is applied to a set of benchmark functions, and the experimental results illustrate its efficiency. Keywords: Particle swarm optimization, differential evolution, numerical optimization. 1 Introduction Particle swarm optimization (PSO) is a novel multi- agent optimization system (MAOS) inspired by social behavior metaphor [12]. Each agent, call particle, flies in a D-dimensional space S according to the historical experiences of its own and its colleagues. The velocity and location for the i th particle is represented as i v r = (v i 1 , …,v id ,…,v i D ) and i x r = ) ,..., ,..., ( D 1 i id i x x x , respectively. Its best previous position is recorded and represented as i p r =(p i 1 , …, p id , …, p i D ), which is also called pbest . The index of the best pbest is represented by the symbol g , and g p r is called gbest. At each step, the particles are manipulated according to the following equations [15]: v id =w·v id +c 1 ·rand()·( p id -x id )+c 2 ·rand()·( p gd -x id ) (1a) x id = x id + v id (1b ) where w is inertia weight, c 1 and c 2 are acceleration constants, rand() are random values between 0 and 1. Several researchers have analyzed it empirically [1, 11, 20] and theoretically [3, 5], which have shown that the particles oscillate in different sinusoidal waves and converging quickly , sometimes prematurely , especially for PSO with small w [20] or constriction coefficient[3]. The concept of a more -or-less permanent social topology is fundamental to PSO [10, 12], which means the pbest and gbest should not be too closed to make some particles inactively [8, 19, 20] in certain stage of evolution. The analysis can be restricted to a single dimension without loss of generality. From equations (1), v id can become small value, but if the | p id -x id | and | p gd -x id | are both small, it cannot back to large value and lost exploration capability in some generations. Such case can be occured even at the early stage for the particle to be the gbest, which the | p id -x id | and | p gd -x id | are zero , and v id will be damped quickly with the ratio w. Of course, the lost of diversity for | p id -p gd | is typically occured in the latter stage of evolution process. To maintain the diversity, the DPSO version [20] introduces random mutations on the x id of particles with small probability c l , which is hard to be determined along with the evolution, at least not be too large to avoid disorganization of the swarm. It can be improved by a bell-shaped mutation, such as Gaussian distribution [8], but a function of consensus on the step-size along with the search process is preferable [11]. A bare bones version [11] for satisfying such requirements is to replace the equations (1) by a Gaussian mutation with the mean (p id +p gd )/2 and the standard deviation | p id -p gd |, which maybe also be inefficient when | p id -p gd | is very small, and is too radically since it turns the PSO into a variaty of in evolution strategies (ES) [2].